Notes on U-tetration ( iteration of x->
exp(x)-1 )

(Last
update at13.Feb.2011 remark: I changed
the use of the symbol U of the previous version to get a clearer distinction
between symbols for the function and that for the matrices)

The basic notation, the formal powerseries

Here I consider the formal powerseries for
iterates of the function d(x) = exp(x)-1 .

I'll use the notations

d(x,
1) = d(x) = exp(x) – 1 d(x, h+1) = d( d(x,1),h) d(x, 0) = x

I call the number-of-iterations
"height" (of the powertower) and use the letter h for that parameter.

For negative integer h we have the iterates of log(1+x):

d(x,–1) = log(1+x) d(x,–2) = log(1+log(1+x)) ...

It is possible to define formal powerseries for
fractional iterations consistent with the addition of (iteration-) heights:

d( d(x, h1), h2) = d(x, h1+h2)

Using a "matrix-operator" (a conveniently scaled
"Bell"-matrix)

A procedure for the computation can be found
for instance in L. Comtet. Comtet introduces the self-composition using the
Newton-binomial-formula for the formal powerseries.

In this article I simply use the
matrix-logarithm of a small modification of the Bell-matrix for the d(x)

Let
U1 the column-vector of coefficients of the formal powerseries for
d(x) U1 =
column(0,1,1/2!,1/3!,…)

Then
let U0 the column-vector for d(x)0 = 1 U0 =
column(1,0,0,0,…)

Let
Uc the column-vector for the c'th power of d(x)c (not the iterate!) Uc = column( )

Then let U
the matrix of concatenation of all Uc-vectors. U
is then the factorially scaled matrix of the Stirling-numbers 2nd
kind, as, for instance defined in [A&S]

Matrixoperator U:

Matrix of Stirlingnumbers 2nd
kind
(the Bell-matrix for d(x))

Further, let's define a general type of
Vandermondevector

V(x)
= row(1, x, x2, x3, …)

With this we can construct the (iterable)
matrix-product:

V(x) * U = V(d(x)) = V(d(x,1)) V(x) * U2 =
V(d(x,1))*U = V(d(x,2))

V(x) * Uh =
V(d(x,h))

This expression is so far only meant as
expression for the formal composition of the powerseries, which we find in the
second column of V(d(x,h)).
This means, to get that powerseries we need only consider the second column of
the h'th
power of U, let's write it this way:

d(x,h) = V(x) * Uh [,1]

Formal powerseries for the fractional iterates d(x,h) ,
h
noninteger

We can get formal powerseries for fractional
iterates analoguosly if we can define fractional powers of the matrix U.

Because the matrix has a unit-diagonal the best
method to compute a fractional power is to use the matrix-logarithm defined by
the mercator-series using U–I
as parameter.

UL = Log( U ) = (U–I) – (U–I)2/2
– (U–I)3/3 + … – …

The argument U–I
is nilpotent to any selected truncation size, so that series gives finite
expressions for the entries of UL
at any finite row/column-index and these evaluate then to exact entries in
rational numbers. This allows a meaningful interpretation of that
matrix-logarithm.

To compute fractional powers of U we can then similarly use the
matrix-exponential

Uh
= Exp( h * Log(U))

And then, as above we get the formal
powerseries for the fractional iterate by the second column of the h'th
power of U:

d(x,h)
= V(x) * Uh [,1] //
also valid for all h

For instance, for h=1/2 we get the matrix U1/2

Matrixoperator U1/2:

giving the formal powerseries

whose coefficients increase strongly such that
the radius of convergence of this series is zero.

From this we can write the two-parameteric
function d(x,h)
formally as matrix-product introducing the matrix POLY
of coefficients of that polynomials

d(x,h)
= V(x) * POLY * V(h) ~

where the top-left truncation of POLY
looks like

Matrix POLY of coefficients for the
two-parametric function d(x,h):

For a given height h this means, that we have to
multiply POLY with the transpose V(h)~ to
get the column-vector U1
corresponding to Uh[,1]
which contains the coefficients for the formal powerseries for the h'th
iterate:

The interesting thing is here, whether we can
change order of summation. This is then interesting, if we always assume x=1
in d(x,h).

Since each column is associated with one power
of the h-parameter,
we may sum up each column to have only one term for a powerseries in h.

We see, that the sequence of entries along a
column diverge after a local minimum and looking at coefficients of higher
index it seems that they will grow at least hypergeometrically (like the
factorials) .

I constructed a Noerlund-means method of
summation which I can apply to the powerseries if I want to estimate the
column-sums meaning we apply x=1 at d(x,h)
keeping h
variable. That summation is parametrized to sum alternating series of order gamma(k)1.5
Remark: this is absolutely experimental and no proof for the correct value ,
for conversion to convergent series and/or regularity of the result is given so
far. But because the results and also the intermediate values seem to be
reasonable I give here the assumed column-sums in the brown row at the bottom
of the table.

This would then give a powerseries for d(1,h)
in terms of h:

d(1,h)
= 1 + 0.4330 h + 0.1758 h2 + 0.06852 h3 + ...

and for h=1 this should be e - 1 ~ 1.718 ... and this value
is good approximated by the row-sum of the last (brown-marked) row.

Checking the convergence of the partial sums for integer
and fractional heights (h=1,2,-1, 0.5)

The same with h-parameter h= -1, so the inverse-iteration;
giving log(1+x)
with x=1:

U -11 = POLY * V(-1)~ Approx = Noerlund(0,0.95) * dV(1)* U-11 // we use only small order of the
Noerlundtransformation to accelerate convergence

U-11

Partial sums: Approx:

0

0

1.00000000000

0.444444444444

-0.500000000000

0.592592592593

0.333333333333

0.658436213992

-0.250000000000

0.680384087791

0.200000000000

0.688797439415

-0.166666666667

0.691601889956

0.142857142857

0.692623801540

-0.125000000000

0.692964438735

0.111111111111

0.693085511172

-0.100000000000

0.693125868651

0.0909090909091

0.693140005390

-0.0833333333333

0.693144717637

0.0769230769231

0.693146352716

-0.0714285714286

0.693146897743

0.0666666666667

0.693147085613

-0.0625000000000

0.693147148237

0.0588235294118

0.693147169718

-0.0555555555556

0.693147176879

0.0526315789474

0.693147179326

-0.0500000000000

0.693147180142

0.0476190476190

0.693147180420

-0.0454545454545

0.693147180513

0.0434782608696

0.693147180544

-0.0416666666667

0.693147180555

0.0400000000000

0.693147180558

-0.0384615384615

0.693147180559

0.0370370370370

0.693147180560

-0.0357142857143

0.693147180560

0.0344827586207

0.693147180560

-0.0333333333333

0.693147180560

The same with h-parameter h= 1/2, so giving the formal
powerseries for the half-iteration, using x=1

The approximation is very difficult now since
we have a divergent series in the terms, whose characteristic is not yet really
known, but seems to be more than hypergeometric, so a high
order for Noerlund-summation is needed. However, for size = 64 I got
an approximation using Eulersummation of order
ord=2.5
which is shown below.

But using more terms it is obvious that that
convergence is only local; So I used an experimental stronger parametrization
of the Noerlund-summation (in my implementation with two parameters),
nonetheless apparently arriving at the same value

Approx = Noerlund(1.4, 1.3) * dV(1)* U0.51 //
we need Noerlund-summation for this because series is strongly diverging

Using 256 terms we find the following approximation :

U0.51

Partial sums: Approx:

0

0

0

1

1.00000000000

0.4166666666666666666666667

2

0.250000000000

0.6802187646414115767312961

3

0.0208333333333

0.8543367076839841850527931

4

1.063167461E-204

0.9728122533370859110058801

5

0.000260416666667

1.055158890718194379537863

6

-0.0000759548611111

1.113322049145836683676599

7

0.00000155009920635

1.154927026644590189403069

8

0.0000154041108631

1.184995099232425260962484

9

-0.00000907453910384

1.206912382008609448210350

10

-0.0000000828199706170

1.223005413341123686437128

11

0.00000360740727676

1.234896974845728865717737

12

-0.00000169514972633

1.243733093659537186974143

13

-0.00000133089916348

1.250331573819650138051143

14

0.00000177521444910

1.255281207630202865239360

15

0.000000370353976658

1.259009183115809527489827

16

-0.00000191475684776

1.261827541981969525892891

17

0.000000344673434042

1.263965589964040800596140

18

0.00000241913411616

1.265592744460861863146730

19

-0.00000147705874041

1.266834787928400056087080

20

-0.00000360462602023

1.267785523004340843822270

21

0.00000426030599723

1.268515189788157734675139

22

0.00000619401781838

1.269076583543275748684661

23

-0.0000126252925336

1.269509526672548941751353

24

-0.0000117360887110

1.269844154811896911030546

25

0.0000413952285774

1.270103343096363738965483

26

0.0000222030302120

1.270304505488932369876442

27

-0.000153108566769

1.270460934623827685972399

28

-0.0000278327871472

1.270582803292161357418455

29

0.000641018661899

1.270677915673107836555729

30

-0.000111307516319

1.270752272718673998185568

31

-0.00303026666274

1.270810498998789451747707

32

0.00167666962999

1.270856165904167737114013

33

0.0160951154545

1.270892037054775901782266

34

-0.0157084159784

1.270920255131584852660284

35

-0.0954804645039

1.270942484470710617007680

36

0.139489606827

1.270960020154843030156470

37

0.628520649401

1.270973871664026359126703

38

-1.27694165810

1.270984827158658745978229

39

-4.55956399021

1.270993502982183114841320

40

12.4027724564

1.271000381858227299430625

41

36.1854546816

1.271005842420900556793057

42

-129.305594720

1.271010182086980195864213

…

…

…

242

1.99391580834E202

1.271027413889951521419677

243

-1.89349634501E202

1.271027413889951521420391

244

-7.29686595931E204

1.271027413889951521421002

245

9.45085000863E204

1.271027413889951521421524

246

2.71412390180E207

1.271027413889951521421972

247

-4.46257691886E207

1.271027413889951521422355

248

-1.02596030513E210

1.271027413889951521422683

249

2.04869812900E210

1.271027413889951521422964

250

3.94079346151E212

1.271027413889951521423204

251

-9.27402089917E212

1.271027413889951521423411

252

-1.53792247833E215

1.271027413889951521423587

253

4.17365586728E215

1.271027413889951521423739

254

6.09718075121E217

1.271027413889951521423868

255

-1.87702245373E218

1.271027413889951521423980

We seem to arrive at a value d( 1, 0.5) ~
1.27102741388995152142.., which reinserted into that powerseries
(and again be Noerlund-summed) gives a reasonable approximation to d(d(1,0.5),0.5) =
d(1,1) .

However, it is possible that this is still only
a local approximation. It seems, that the growthrate of the terms is even more
than hypergeometric; a plot of the quotients of successive coefficients
indicates strongly, that in the average the coefficients grow dependent on the
index k
like O((k!)1+δ)
(with oscillations) where if δ>0 significantly the used Noerlund-summation
is not strong enough. The following plot shows the growthrate by the formula

Note: I used dxp(x,h) in the program for the drawing of the graphic instead
of d(x,h)